1 | initial version |

Suppose `n(x)`

was defined somehow (in fact, in extenso) but see also my related question about how to import Maxima built-in/package functions. My current result so far:

```
absm(p)=integral(abs(x)^p*n(x), x, -infinity, infinity)
absm
p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
a(k,p)=integral(integral(abs(u)^p*abs(u+sqrt(k-1)*v)^p*n(v)*n(u), v, -infinity, infinity), u, -infinity, infinity)
a(k,p)
1/2*integrate(abs(u)^p*integrate(abs(sqrt(k - 1)*v + u)^p*e^(-1/2*u^2 - 1/2*v^2), v, -Infinity, +Infinity), u, -Infinity, +Infinity)/pi
ab(p)=a(1,p/2).simplify()
ab
p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
bool(absm == ab)
True
```

This is a partial result but `a(k,p)`

can't be simplified in terms of gamma's or whatnots.

2 | No.2 Revision |

Suppose `n(x)`

was defined somehow (in fact, in extenso) but see also my related question about how to import Maxima built-in/package functions. My current result so far:

```
absm(p)=integral(abs(x)^p*n(x), x, -infinity, infinity)
absm
p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
a(k,p)=integral(integral(abs(u)^p*abs(u+sqrt(k-1)*v)^p*n(v)*n(u), v, -infinity, infinity), u, -infinity, infinity)
a(k,p)
1/2*integrate(abs(u)^p*integrate(abs(sqrt(k - 1)*v + u)^p*e^(-1/2*u^2 - 1/2*v^2), v, -Infinity, +Infinity), u, -Infinity, +Infinity)/pi
ab(p)=a(1,p/2).simplify()
ab
p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
bool(absm == ab)
True
```

This is a partial result but `a(k,p)`

can't be simplified in terms of gamma's or whatnots.

How do you make changes of variables here?

3 | No.3 Revision |

Suppose `n(x)`

was defined somehow (in fact, in extenso) but see also my related question about how to import Maxima built-in/package functions. My current result so far:

```
absm(p)=integral(abs(x)^p*n(x), x, -infinity, infinity)
absm
p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
a(k,p)=integral(integral(abs(u)^p*abs(u+sqrt(k-1)*v)^p*n(v)*n(u), v, -infinity, infinity), u, -infinity, infinity)
a(k,p)
1/2*integrate(abs(u)^p*integrate(abs(sqrt(k - 1)*v + u)^p*e^(-1/2*u^2 - 1/2*v^2), v, -Infinity, +Infinity), u, -Infinity, +Infinity)/pi
ab(p)=a(1,p/2).simplify()
ab
p |--> 2^(1/2*p)*gamma(1/2*p + 1/2)/sqrt(pi)
bool(absm == ab)
True
```

This is a partial result but `a(k,p)`

can't be simplified in terms of gamma's or whatnots.

How do you make changes of variables ~~here?~~in Sage?

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